Re: Dedekind Cuts, Fundamental Sequences: why?

On Thu, 07 Jun 2007 07:54:50 -0400, Hatto von Aquitanien wrote:
Virgil wrote:

In article <l4adnYr9ws-7PvrbnZ2dnUVZ_vyunZ2d@xxxxxxxxxxxxx>,
Hatto von Aquitanien <abbot@xxxxxxxxxxxxxx> wrote:

Dave Seaman wrote:
9. It remains to prove that R is complete. Let A = {A_i:i
in I} be a nonempty indexed subset of R = C/I that is bounded
above by a real number b.

It is not clear if you intend by "real number" an equivalence class in C,
or something else defined prior to this development.

Each real is a distinct equivalence class.

I was just trying to clarify that there is nothing being introduced
from "outside".

Let {b_n} be a representative
of the equivalence class for b. For each a_i in A, we can
choose a representative Cauchy sequence {a_{i,n}} such that
a_{i,n} <= b_n for each i in I and each n > 0. We now need
to produce a sequence {d_n} that will represent a least
upper bound for A. For each n, we can choose d_n such that
d_n <= b_n and also such that 0 <= inf({|d_n-a_{i,n}|: i in I})
< 1/n. Then the sequence {d_n} is the required representative
of a lub for A.

I have to think about this more. There are many assumptions in this last
step which I have not given much thought to in the past. If I am not
mistaken you have what might be called second order Cauchy sequences.
That is Cauchy sequences of Cauchy sequences.

Each equivalence class contains an infinity of Cauchy sequences of
rationals.

Is it possible that the elements of d_n might be irrational?

No, each d_n is chosen to be rational.

To show that a particular real exists, one only need show
that an appropriate rational Cauchy sequence exists.

"Let A = {A_i:i in I} be a nonempty indexed subset of R = C/I that is
bounded above by a real number b."

The members of R, and therefore of A are equivalence classes of Cauchy
sequence.

"For each a_i in A, we can choose a representative Cauchy sequence {a_{i,n}}
such that a_{i,n} <= b_n for each i in I and each n > 0."

As I look at this again, I realize I do not understand it.
Abbreviate "representative Cauchy sequence" as rcs. Then for each A_i we
will have

b_1=>{a_{i,1}}=rcs_i1
b_2=>{a_{i,2}}=rcs_i2
b_3=>{a_{i,3}}=rcs_i3

etc.,

or

rcs_11, rcs_21, rcs_31, etc., such that rcs_i1<=b_1
rcs_12, rcs_22, rcs_32, etc., such that rcs_i2<=b_2
rcs_13, rcs_23, rcs_33, etc., such that rcs_i3<=b_3

etc.

Might the rcs_in be irrational?

We can always choose them to be rational.

There is no problem in software design which cannot be solved by
introducing
another level of indirection. In this case there appears to be
self-referential recursion going on.


There is a layered construction going on.

The base layer is the arithmetic of (positive) naturals.
The next layer builds, say, the positive rationals from the naturals.
The next layer adds on the negative rationals.
The top layer, so far, builds the reals from the rationals.

I'm not sure if the development above implies that d_n will be a Cauchy
sequence of, in general, both rational and irrational elements. I do,
however have a book which discusses Cauchy sequences of real numbers. Such
a sequence would be a sequence of sequences of rational numbers which do
not necessarily converge to rational numbers. That is what I meant by
second order sequences.

The d_n are always rational in step 9. Once we have finished steps 1-9,
we can then go on to prove theorems about the constructed field R, one of
which says that every Cauchy sequence in R converges in R. That's where
we begin to encounter sequences of sequences.

An common alternate sequence, though a bit less elegant, goes from
naturals to integers to rationals to reals,

That is, the structure refers to
itself from a level of construction above the level being referred to.

Such references are always to the structure "below" the one being
defined.

There is something called the Xerox effect which says that copies of
copies of copies, ..., are never as good as the original.

Each of the "copies" here contains an exact image of its source within
it, but contains more.


From here on, I am going to drop the references to equivalence classes and
treat the Cauchy sequence as the same as the equivalence class of Cauchy
sequences

What I was trying to suggest - though I don't know that I would try to
sustain the argument - is that if one wanted to claim that there was some
imperfection in the Cauchy sequence representation of a real number, then
such imperfections would be compounded when constructing sequences of
sequences. For example, the rational numbers are represented by a subset
of the Cauchy sequence representing the real numbers. I don't see a way to
argue that these could be less than perfect copies. On the other hand,
someone might try to object that the fidelity of representation of a real
using a sequence of rational numbers is less than perfect.

Weyl mistrusted the representation of real numbers as either Cauchy
sequences or Dedekind cuts, arguing that no proof had been given that these
sets actually 'inherit' the properties of their members. The more I look
into things, the more I realize, he saw the whole notion of real numbers as
they are defined to be specious.

Perhaps no proof of that fact was included in in the argument he saw, but
that's because it's the sort of thing that any competent reader is
expected to be able to supply for himself.

It seems as though it would take a
considerable effort to grasp what he is really trying to argue, and I'm not
sure the reward would be great. But if I were to side with Weyl and argue
that the step going from rational numbers to sets of rational numbers is
not supported by necessary proof, it seems reasonable that the second step
of going from Cauchy sequences of rational numbers to Cauchy sequences of
Cauchy sequences of rational numbers was also in need of proof.

Of course it needs proof. The proof is elementary.

The entity I am calling sqrt(2) was specifically rejected for membership
in the set used to define the Cauchy sequences, the equivalence classes
of which represent the real numbers.

Given any positive rational, a_0, the recursion formula
a_(n+1) = (a_n^2 + 2)/(2*a_n) generates a Cauchy sequence of rationals
which "converges" to sqrt(2).

It arose from the algebra of the field
of rational numbers. It may seem ridiculous to raise this objection, but
it seems to me worth considering. I can now talk about {x:2=x^2, x \in
\R}. Is that the same {x:2=x^2, x \in \Q}?

No. The former set contains two members, the latter none.

So the hole I was told about seems to have been a manifestation of prior
experience with the finished domain. If that example had played any
essential part in the development, then we would have a compromised
structure.

Do points (1)-(9) help?

They help clarify what is going on. Unfortunately, I believe this is one
of those cases where we either accept the identity of entities at
different
levels of abstraction, or fall into infinite regress. There has to be a
Cauchy sequence of representative Cauchy sequences of Cauchy sequences
hiding around the corner.

Not when you finally sort things out properly.

I guess one could argue that even if we do have Cauchy sequences in which
some of the representations have irrational elements, they must all have
representations which are purely rational due to the fact that the real
numbers were completely defined by the latter. That is, of course,
assuming that such a construction actually took place.

Yes, a sequence of reals has a representation as a sequence of sequences
of rationals.

--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
.

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